Coin Toss Probability Setup
Let's break this down step by step.
1. Probability Space:
A probability space consists of three main components:
- A sample space ,
- A sigma-algebra ,
- A probability measure .
For a biased coin, the sample space consists of the possible outcomes of a single coin toss. So, where is heads and is tails.
Given that we're tossing the coin times, the sample space for tosses would be the set of all -length sequences of and . So, the sample space for tosses is:
The sigma-algebra on is the power set of , i.e., the set of all possible subsets of . Since is finite (with only 2 elements), this is trivial. However, for tosses, would be the power set of .
Lastly, is the probability measure. Since the coin is biased, let be the probability that the coin shows heads and be the probability that it shows tails. Therefore, for a single toss: For tosses, the probability of a specific sequence is the product of the probabilities of each toss outcome. For instance, the probability of the sequence in 3 tosses would be .
2. Random Variable for the -th toss:
A random variable is a function that assigns a real number to each outcome in a sample space.
Let's define to be the random variable that gives the result of the -th toss. For any sequence in , the value of the random variable at is .
Formally, for in , for .
It's worth noting that if you'd like the random variable to map to real numbers (which is often the convention), you could define such that:
Where 1 represents heads and 0 represents tails.
Given a biased coin with the probability of heads being and tails being , let's define the probability measure on the sample space for coin tosses.
For any specific sequence in , where each can be either (heads) or (tails), the probability measure is given by:
Or, in a more concise mathematical notation:
where is the number of occurrences of in the sequence .
Given that is the random variable representing the outcome of the -th toss, where:
Then, the sum represents the total number of heads in coin tosses. This sum will follow a binomial distribution.
The binomial distribution gives the probability of having successes (in this context, heads) in independent Bernoulli trials (coin tosses), each with the same probability of success.
Thus, the distribution of is , and the probability mass function (PMF) of is given by:
- is the binomial coefficient, representing the number of ways to choose successes from trials.
- is the probability of getting heads.
- is the probability of getting tails.
Here, can range from 0 to , inclusive.
Thank you! If you have any more questions or need further clarification on any topic, feel free to ask. Have a great day!